By John C. Baez, Danny Stevenson (auth.), Nils Baas, Eric M. Friedlander, Björn Jahren, Paul Arne Østvær (eds.)

The 2007 Abel Symposium came about on the college of Oslo in August 2007. The aim of the symposium used to be to compile mathematicians whose examine efforts have resulted in contemporary advances in algebraic geometry, algebraic K-theory, algebraic topology, and mathematical physics. a standard subject matter of this symposium was once the advance of recent views and new structures with a express style. because the lectures on the symposium and the papers of this quantity exhibit, those views and structures have enabled a broadening of vistas, a synergy among once-differentiated topics, and recommendations to mathematical difficulties either outdated and new.

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Cat. 12 (2004), 423–491. Also available as arXiv:math/0307200. 4. J. C. Baez and U. Schreiber, Higher gauge theory, in Categories in Algebra, Geometry and Mathematical Physics, eds. A. , Contemp. Math. 431, AMS, Providence, RI, 2007, pp. 7–30. Also available as arXiv:math/0511710. 5. T. Bartels, Higher gauge theory I: 2-bundles. Available as arXiv:math/0410328. 6. J. Braho, Haefliger structures and linear homotopy (appendix), Trans. Amer. Math. Soc. 282 (1984), 529–538. 7. L. Breen, Bitorseurs et cohomologie non-ab´elienne, in The Grothendieck Festschrift I, eds.

Then Ü and Ý belong to the À same fiber of À over À , so Ý ½ Ü ¾ À . We set «º Ü Ý» ºØºÝ ½ Ü» Ý» A straightforward calculation shows that « is well defined and that « and ¬ are À inverse to one another. To conclude, we need to show that G is a well-pointed topological group. For this it is sufficient to show that G is a “proper” simplicial space in the sense of May [23] (note that we can replace his “strong” NDR pairs with NDR pairs). 3 that G Ô G is an NDR pair for all . In particular G ¼ G is an NDR pair.

It is then Ä½ easy to see that the induced map Ä ½ is a bijection. 5 Proof of Lemma 3 Suppose that ½ G¼ G½ Ô ½ G¾ is a short exact sequence of topological 2-groups, so that we have a short exact sequence of topological crossed modules: ½ ½ À¼ À½ Ø¼ Ø½ ¼ Ô À¾ ½ Ø¾ ½ Ô ¾ ½ The Classifying Space of a Topological 2-Group Also suppose that U representing a class in 27 Í is a good cover of Å , and that º ÀÄ ½ ºU G½». We claim that the image of ½ ÀÄ ½ºÅ G½ » £ Ï ÀÄ ºÅ G¼ » equals the kernel of » is a cocycle Ô£Ï ÀÄ ½ºÅ G½ » ÀÄ ½ºÅ G¾ » Ô If the class is in the image of £ , it is clearly in the kernel of £ .